I'm watching this exchange with interest but I think I'm missing
something.
Torkel Franzen <torkel at sm.luth.se> wrote:
>So let us consider the suggested justification, taking phi in
>>(1) Prov_S(phi)->phi
>>to be a logical contradiction psi&~psi. (1) then states that S is
>consistent, Con_S.
First of all, in a certain sense, (1) doesn't "directly state" that S
is consistent. A more "direct" reading of (1) is, "if S proves that
psi both holds and does not hold, then psi both holds and does not hold."
(We could quibble further and say that a "direct" reading of (1) is a
monstrously cumbersome statement in *arithmetic*, not even a direct
statement about *syntax*, but let me try to avoid going down that path.)
It is then a trivial *consequence* of (1) that S is consistent. However,
if we're trying to figure out whether we can justify (1), it doesn't
follow that we have to figure out a "direct" justification that "S is
consistent"; it suffices to figure out a "direct" justification of what
I call the "direct" reading of (1), and let someone else turn the wheels
and grind out "S is consistent" as a consequence afterwards.
Neil Tennant appears to be arguing that if I *really* am willing to assert
*any* sentence phi for which I have an S-proof, then that means that I
would even be willing to assert psi & ~psi if I had an S-proof of it. And
I don't see anything incomprehensibly baffling about such a claim; it
could for example be interpreted as saying that my loyalty to S is so
strong that it exceeds my commitment to avoid asserting contradictions.
Why is this baffling? What am I missing? Maybe a distinction between
a counterfactual conditional ("If S were to prove a contradiction...")
and a material conditional ("If S proves a contradiction...")?
Tim